def test_poly_gcdex():
f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
g = x**3 + x**2 - 4*x - 4
assert poly_half_gcdex(f, g, x) == \
(Poly(-x/5+Rational(3,5), x), Poly(x+1, x))
assert poly_half_gcdex(Poly(f, x), Poly(g, x)) == \
(Poly(-x/5+Rational(3,5), x), Poly(x+1, x))
assert poly_gcdex(f, g, x) == \
(Poly(-x/5+Rational(3,5), x), Poly(x**2/5-6*x/5+2, x), Poly(x+1, x))
assert poly_gcdex(Poly(f, x), Poly(g, x)) == \
(Poly(-x/5+Rational(3,5), x), Poly(x**2/5-6*x/5+2, x), Poly(x+1, x))
f = x**4 + 4*x**3 - x + 1
g = x**3 - x + 1
s, t, h = poly_gcdex(f, g, x)
S, T, H = poly_gcdex(g, f, x)
assert s*f + t*g == h
assert S*g + T*f == H
assert poly_gcdex(2*x, x**2-16, x) == \
(Poly(x/32, x), Poly(-Rational(1,16), x), Poly(1, x))
def test_poly_gcdex():
f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
g = x**3 + x**2 - 4*x - 4
assert poly_half_gcdex(f, g, x) == \
(Poly(-x/5+Rational(3,5), x), Poly(x+1, x))
assert poly_half_gcdex(Poly(f, x), Poly(g, x)) == \
(Poly(-x/5+Rational(3,5), x), Poly(x+1, x))
assert poly_gcdex(f, g, x) == \
(Poly(-x/5+Rational(3,5), x), Poly(x**2/5-6*x/5+2, x), Poly(x+1, x))
assert poly_gcdex(Poly(f, x), Poly(g, x)) == \
(Poly(-x/5+Rational(3,5), x), Poly(x**2/5-6*x/5+2, x), Poly(x+1, x))
f = x**4 + 4*x**3 - x + 1
g = x**3 - x + 1
s, t, h = poly_gcdex(f, g, x)
S, T, H = poly_gcdex(g, f, x)
assert s*f + t*g == h
assert S*g + T*f == H
assert poly_gcdex(2*x, x**2-16, x) == \
(Poly(x/32, x), Poly(-Rational(1,16), x), Poly(1, x))
def test_poly_gcdex():
f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
g = x**3 + x**2 - 4*x - 4
assert poly_half_gcdex(f, g, x) == \
(Poly(-x+3, x), Poly(5*x+5, x))
assert poly_half_gcdex(Poly(f, x), Poly(g, x)) == \
(Poly(-x+3, x), Poly(5*x+5, x))
assert poly_gcdex(f, g, x) == \
(Poly(-x+3, x), Poly(x**2-6*x+10, x), Poly(5*x+5, x))
assert poly_gcdex(Poly(f, x), Poly(g, x)) == \
(Poly(-x+3, x), Poly(x**2-6*x+10, x), Poly(5*x+5, x))
f = x**4 + 4*x**3 - x + 1
g = x**3 - x + 1
s, t, h = poly_gcdex(f, g, x)
S, T, H = poly_gcdex(g, f, x)
assert s*f + t*g == h
assert S*g + T*f == H
def test_poly_gcdex():
f = x**4 - 2 * x**3 - 6 * x**2 + 12 * x + 15
g = x**3 + x**2 - 4 * x - 4
assert poly_half_gcdex(f, g, x) == \
(Poly(-x+3, x), Poly(5*x+5, x))
assert poly_half_gcdex(Poly(f, x), Poly(g, x)) == \
(Poly(-x+3, x), Poly(5*x+5, x))
assert poly_gcdex(f, g, x) == \
(Poly(-x+3, x), Poly(x**2-6*x+10, x), Poly(5*x+5, x))
assert poly_gcdex(Poly(f, x), Poly(g, x)) == \
(Poly(-x+3, x), Poly(x**2-6*x+10, x), Poly(5*x+5, x))
f = x**4 + 4 * x**3 - x + 1
g = x**3 - x + 1
s, t, h = poly_gcdex(f, g, x)
S, T, H = poly_gcdex(g, f, x)
assert s * f + t * g == h
assert S * g + T * f == H
def log_to_atan(f, g):
"""Convert complex logarithms to real arctangents.
Given a real field K and polynomials f and g in K[x], with g != 0,
returns a sum h of arctangents of polynomials in K[x], such that:
df d f + I g
-- = -- I log( ------- )
dx dx f - I g
"""
if f.degree < g.degree:
f, g = -g, f
p, q = poly_div(f, g)
if q.is_zero:
return 2*atan(p.as_basic())
else:
s, t, h = poly_gcdex(g, -f)
A = 2*atan(quo(f*s+g*t, h))
return A + log_to_atan(s, t)
def log_to_atan(f, g):
"""Convert complex logarithms to real arctangents.
Given a real field K and polynomials f and g in K[x], with g != 0,
returns a sum h of arctangents of polynomials in K[x], such that:
df d f + I g
-- = -- I log( ------- )
dx dx f - I g
"""
if f.degree < g.degree:
f, g = -g, f
p, q = poly_div(f, g)
if q.is_zero:
return 2*atan(p.as_basic())
else:
s, t, h = poly_gcdex(g, -f)
A = 2*atan(quo(f*s+g*t, h))
return A + log_to_atan(s, t)